Answer:
14 ft × 14 ft × 8.75 ft
Step-by-step explanation:
A garbage company is designing an open rectangular container that should have a volume of 1,715 cubic feet.
So we have the length of the container = "x" ft, the width of the container = "y" ft and the height of the container = "z" ft
Therefore the volume of the rectangular container would be:
x * y * z = 1715 ft³
z = 1715 / x * y
The cost of making the bottom of the container is $ 5 per square foot, that is:
5 * (x * y)
Now, area of all sides of the container would be:
2 * (x * z + y * z) = 2 * z * (x + y)
We know that it has been given that the cost of making all the sides of the container is = $ 4 per square foot, so:
4 * (2 * z * (x + y)) = 8 * z * (x + y)
In total the costs would be:
5 * (x * y) + 8 * z * (x + y)
If we replace z, in the previous equation we have:
5 * (x * y) + 8 * (1715 / x * y) * (x + y)
solving, and we would have that the total cost would be:
C = 5 * (x * y) + 13720 / x + 13720 / y
Now we will find the derivative of C and make it equal to zero:
dC / dx = 0; dC / dy = 0
For dC / dx = 0:
dC / dx = 5 * y + 13720 * -1 / (y ^ 2) + 13720 * 0
0 = 5 * y - 13720 / y ^ 2
5 * y = 13720 / y ^ 2
y ^ 3 = 13720/5 = 2744
y = 14
For dC / dy = 0:
dC / dy = 5 * x + 13720 * 0 + 13720 * -1 / (x ^ 2)
0 = 5 * x - 13720 / x ^ 2
5 * x = 13720 / x ^ 2
x ^ 3 = 13720/5 = 2744
x = 14
now for z:
z = 1715 / (14 * 14)
z = 8.75
Therefore, the dimensions of the container should be 14 ft × 14 ft × 8.75 ft to minimize manufacturing cost.
